Positivity of the effective range for finite range attractive potentials with a repulsive core

TL;DR

This paper proves that for finite-range potentials with a repulsive core and attractive tail, the effective range is always positive if the scattering length exceeds the potential's range, impacting hadron structure interpretations.

Contribution

It provides a rigorous proof that the effective range is positive under specific conditions for finite-range potentials with a repulsive core.

Findings

Effective range remains positive if scattering length > potential range.

The result constrains interpretations of hadronic structures based on effective range signs.

Provides fundamental constraints on the sign of the effective range for certain potentials.

Abstract

In the phenomenological study of exotic hadrons, the sign of the effective range, $r_0$, is invoked as a criterion to distinguish between compact multiquark configurations (associated with $r_0 < 0$) and loosely bound hadronic molecules ($r_0 > 0$). Motivated by this, we investigate the fundamental constraints on the sign of the effective range for single-channel local interactions. We rigorously prove that for finite-range potentials, characterized by an inner repulsive core and an outer attractive tail, the effective range remains strictly positive provided that the scattering length is greater than the range of the potential ($a > R$).